Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “local density approximation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 505 records · Page 28
All sourcesNASA NTRSDOE OSTI
NASA NTRS1994

Absorption in X-ray spectra of high-redshift quasars

We present evidence that X-ray absorption is common in high-redshift quasars. We have studied six high-redshift (z approximately 3) quasars with the ROSAT Position Sensitive Proportional Counter (PSPC) of which four are in directions of low Galactic N(sub H). Three out of these four show excess absorption, while only three in approximately 50 z approximately less than 0.4 quasars do, indicating that such absorption must be common, but not ubiquitous, at high redshifts, and that the absorbers must lie at z greater than 0.4. The six quasars were: S5 0014+81, Q0420-388, PKS 0438-436, S4 0636+680. PKS 2000-330, PKS 2126-158, which have redshifts between 2.85 and 3.78. PKS 0438-436 and PKS 2126-158 show evidence for absorption above the local Galactic value at better than 99.999% confidence level. If the absorber is at the redshift of the quasar, then values of N(sub H) = (0.86(+0.49, -0.28)) x 10(exp 22) atoms/sq cm for PKS 0438-436, and N(sub H) = (1.45(+1.20, -0.64)) x 10(exp 22) atoms/ sq cm for PKS 2126-158, are implied, assuming solar abundances. The spectrum of S4 0636+680 also suggests the presence of a similarly large absorption column density at the 98% confidence level. This absorption reverses the trend for the most luminous active galactic nuclei (AGN) to have the least X-ray absorption, so a new mechanism is likely to be responsible. Intervening absorption due to damped Lyman(alpha) systems is a plausible cause. We also suggest, as an intrinsic model, that intracluster material, e.g., a cooling flow, around the quasar could account for both the X-ray spectrum and other properties of these quasars. All the quasars are radio-loud and three are gigahertz peaked (two of the three showing absorption). No excess absorption above the Galactic value is seen toward Q0420-388. This quasar has two damped Lyman(alpha) systems at z = 3.08. The limit on the X-ray column density implies a low ionization fraction, N(H I)/N(H) approximately greater than 4 x 10(exp -3) (3 (sigma)), for solar abundances, for these systems, and can set a weak limit on the size of the absorber. In the emitted frame these PSPC spectra cover the band approximately 0.5-10 keV, which has been well observed for low-redshift quasars and AGN. Comparison of high and low-redshift spectra in this emitted band shows no change of mean spectral index greater than Delta alpha(sub E) greater than 0.3 (99% confidence) with either redshift or luminosity, for radio-loud quasars.

Elvis, Martin↗
DOE OSTI2025

Predicting interface structure using the minima hopping method

Here, we adapt the minima hopping method (MHM) to the problem of interfacial structure prediction and apply it to study a canonical problem, the tilt grain boundaries in SrTiO 3 . Our method employs a hybrid approach by first exploring the potential energy surface (PES) of different grain boundary samplings with an empirical force field, among which the fifteen candidates with lower energies are then refined using ab initio density functional theory (DFT) calculations. During the exploratory stage, we bias the search using a local order parameter to primarily sample various reconstructions in the vicinity of the interface, while preserving the crystallinity of the bulk regions. We further enhance the search by incorporating initial structures with rigid body displacements to account for translational variations between bulk phases, enabling the MHM to effectively generate both stoichiometric and nonstoichiometric SrTiO 3 Σ⁢3(111)[110] and Σ⁢3(112)[110] grain boundaries. From an algorithmic standpoint, MHM outperforms earlier studies based on genetic algorithms (GA) by identifying more stable interfacial structures of several SrTiO 3 grain boundaries. The performance of the present implementation of the MHM approach is primarily limited by exploring an approximate description of the PES with a rather simple Buckingham potential. This limitation leads to variations in performance when compared to approaches utilizing more advanced surrogate PES models, such as direct DFT-PES sampling or GA with the embedded atom method (EAM). Despite the present limitations, the MHM approach is able to yield interfacial structures with comparable or lower interfacial energies in specific cases, such as Σ⁢3(111)[110] Γ=1, ±0.5 and Σ⁢3(112)[110] Γ= ±1, −2, underscoring the robustness of the MHM approach even with a simple approximation of the DFT PES. The MHM interfacial structure prediction method thus offers an efficient approach to understanding the grain boundaries and heterointerfaces at the atomic scale, providing an important prerequisite for effective materials design.

density functional theory↗
DOE OSTI2023

Energy Frontier Research Centers: Center for the Computational Design of Functional Layered Materials (CCDM) August 1, 2014 - July 31, 2018; Center for Complex Materials from First Principles (CCM) August 1, 2018 - July 31, 2021 (Final Report)

The mission of the DOE Energy Frontier Research Centers CCDM (2014-2018) and CCM (2018-2021) was to theoretically develop, computationally apply, and experimentally validate electronic structure methods for all materials, with a focus on the complex materials, especially layered and two-dimensional materials, strongly-correlated materials, and liquid water. This was achieved by over 200 published journal articles authored by about 17 senior investigators from physics and chemistry and from theory, computation, and experiment, plus their collaborators. In particular, the Centers confirmed the predictive power of the SCAN (strongly constrained and appropriately normed) density functional, which was constructed to satisfy 17 known exact constraints and several appropriate norms. Without being fitted to real bonded systems, and at a modest computational cost, SCAN correctly predicted covalent, ionic, metallic, hydrogen, and van der Waals bonds in many challenging materials. SCAN gave an improved description of defects in semiconductors, surface properties of metals, seven phases of ice, liquid water, liquid and supercooled silicon, subtle structural distortions in ferroelectrics, formation energies and structural predictions for solids, and critical pressures for structural phase transitions. Perhaps most remarkably, SCAN correctly described some strongly-correlated materials that were previously believed to be beyond the reach of density-functional approximations. SCAN is the only density functional that correctly predicts the band gap closing under chemical doping of the cuprate high-temperature superconducting materials. SCAN also predicts a landscape of competing stripe and magnetic phases in the cuprates. For some materials with some codes, SCAN has convergence problems that are greatly reduced by the CCM-developed r 2 SCAN, without loss of accuracy or rigor. SCAN and r 2 SCAN still make some self-interaction error, which is greatly reduced by the CCDM/ CCM-developed local orbital scaling correction (LOSC). These Centers further proved that the fundamental energy gaps of a solid from an orbital energy difference and from total energy differences are the same for a large class of generalized Kohn-Sham (GKS) functionals, including SCAN and standard hybrid functionals, and that symmetry breaking arises when a dynamic density fluctuation drops to zero frequency. The Centers identified new mechanisms for catalysis in layered materials with ions intercalated between the layers, investigated charge density waves both in model systems and in real layered materials, studied changes of band gap with the number of layers, and explored topological ultrathin films, bent nanoribbons, and defects.

08 HYDROGEN↗
DOE OSTI2020

Eliashberg theory of phonon-mediated superconductivity — When it is valid and how it breaks down

We analyze the validity of Eliashberg theory of phonon-mediated superconductivity in 2D systems in light of recent extensive Monte-Carlo studies of the Holstein model. Conventional wisdom says that Eliashberg theory is applicable as long as vertex corrections remain small. For small ratio of the phonon energy Ω 0 and the Fermi energy E F , this condition is supposed to hold even when the dimensionless electron–phonon coupling λ is larger than one, i.e., in the strong coupling regime. A comparison between various quantities computed in the Migdal approximation and those computed by Quantum Monte Carlo prove that this belief is wrong, and we identify analytically some of the ways in which this breakdown occurs for various ‘‘normal state’’ properties at λ = λ cr , where λ cr = O(1). Here, the breakdown occurs at temperatures high enough that neither superconducting nor charge–density wave correlations extend over any significant range of distances, so it cannot be associated with the onset of an instability toward any of the relevant ordered ground-states — rather it is associated with the local physics of classical bipolaron formation. Still, we show that certain properties, including the superconducting Tc and the superconducting gap structure below T c , can be accurately inferred from the strong-coupling limit of Eliashberg theory at λ ≤ λ cr .

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

Exact constraints and appropriate norms in machine-learned exchange-correlation functionals

Machine learning techniques have received growing attention as an alternative strategy for developing general-purpose density functional approximations, augmenting the historically successful approach of human-designed functionals derived to obey mathematical constraints known for the exact exchange-correlation functional. More recently, efforts have been made to reconcile the two techniques, integrating machine learning and exact-constraint satisfaction. We continue this integrated approach, designing a deep neural network that exploits the exact constraint and appropriate norm philosophy to de-orbitalize the strongly constrained and appropriately normed (SCAN) functional. The deep neural network is trained to replicate the SCAN functional from only electron density and local derivative information, avoiding the use of the orbital-dependent kinetic energy density. The performance and transferability of the machine-learned functional are demonstrated for molecular and periodic systems.

Artificial neural networks↗
NASA NTRS1978

Convective heat flux in a laser-heated thruster

An analysis is performed to estimate the convective heating to the wall in a laser-heated thruster on the basis of a solution of the laminar boundary-layer equations with variable transport properties. A local similiarity approximation is used, and it is assumed that the gas phase is in equilibrium. For the thruster described by Wu (1976), the temperature and pressure distributions along the nozzle are obtained from the core calculation. The similarity solutions and heat flux are obtained from the freestream conditions of the boundary layer, in order to determine if it is necessary to couple the boundary losses directly to the core calculation. In addition, the effects of mass injection on the convective heat transfer across the boundary layer with large density-viscosity product gradient are examined.

Wu, P. K. S.↗
NASA NTRS1997

Io's Sodium Corona and Spatially Extended Cloud: A Consistent Flux Speed Distribution

For Io neutral cloud calculations, an SO2 source strength of approximately 4x10(exp 27) molecules/sec was determined by successfully matching the SO2(+) density profile near the satellite deduced from magnetometer data acquired by the Galileo spacecraft during its close flyby on December 7, 1995. The incomplete collision source velocity distribution for SO2 is the same as recently determined for the trace species atomic sodium by Smyth and Combi (1997). Estimates for the total energy loss rate (i.e. power) of O and S atoms escaping Io were also determined and imply a significant pickup current and a significant reduction in the local planetary magnetic field near Io.

Smyth, William H.↗
NASA NTRS2023

Formation of the Lyman Continuum during Solar Flares

The Lyman continuum (LyC; <911.12 Å) forms at the top of the chromosphere in the quiet Sun, making LyC a powerful tool for probing the chromospheric plasma during solar flares. To understand the effects of nonthermal energy deposition in the chromosphere during flares, we analyzed LyC profiles from a grid of field-aligned radiative-hydrodynamic models generated using the RADYN code as part of the F-CHROMA project. The spectral response of LyC, the temporal evolution of the departure coefficient of hydrogen, b 1 , and the color temperature, T c , in response to a range of nonthermal electron distribution functions, were investigated. The LyC intensity was seen to increase by 4–5.5 orders of magnitude during solar flares, responding most strongly to the nonthermal electron flux of the beam. Generally, b 1 decreased from 10 2 –10 3 to closer to unity during solar flares, indicating a stronger coupling to local conditions, while T c increased from 8–9 to 10–16 kK. T c was found to be approximately equal to the electron temperature of the plasma when b1 was at a minimum. Both optically thick and optically thin components of LyC were found to be in agreement with the interpretation of recent observations. The optically thick layer forms deeper in the chromosphere during a flare compared to quiescent periods, whereas the optically thin layers form at higher altitudes due to chromospheric evaporation, in low-temperature, high-density regions propagating upward. We put these results in the context of current and future missions.

Solar flares↗
DOE OSTI2021

Visualization of Plasmonic Couplings Using Ultrafast Electron Microscopy

Hybrids of graphene and metal plasmonic nanostructures are promising building blocks for applications in optoelectronics, surface-enhanced scattering, biosensing, and quantum information. An understanding of the coupling mechanism in these hybrid systems is of vital importance to its applications. Previous efforts in this field mainly focused on spectroscopic studies of strong coupling within the hybrids with no spatial resolution. In this study, we report direct imaging of the local plasmonic coupling between single Au nanocapsules and graphene step edges at the nanometer scale by photon-induced near-field electron microscopy in an ultrafast electron microscope for the first time. The proximity of a step in the graphene to the nanocapsule causes asymmetric surface charge density at the ends of the nanocapsules. Computational electromagnetic simulations confirm the experimental observations. The results reported here indicate that this hybrid system could be used to manipulate the localized electromagnetic field on the nanoscale, enabling promising future plasmonic devices.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2024

Diffusion quantum Monte Carlo approach to the polaritonic ground state

Making and using polaritonic states (i.e., hybrid electron-photon states) for chemical applications has recently become one of the most prominent and active fields that connects the communities of chemistry and quantum optics. Modeling of such polaritonic phenomena using ab initio approaches calls for new methodologies, leading to the reinvention of many commonly used electronic structure methods, such as Hartree-Fock, density functional, and coupled cluster theories. Here in this work, we explore the formally exact diffusion quantum Monte Carlo approach to obtain numerical solutions to the polaritonic ground state during the dissociation of the H 2 molecular system. Here, we examine various electron-nuclear-photon properties throughout the dissociation, such as changes to the minimum of the cavity Born-Oppenheimer surface, the localization of the electronic wave function, and the average mode occupation. Finally, we directly compare our results to that obtained with state-of-the-art, yet approximate, polaritonic coupled cluster approaches.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2025

Simulating large one-dimensional neutral-atom quantum systems

While abstract models of quantum computation assume a closed system of two-level states, practical quantum devices inevitably couple to the environment in some way, creating sources of noise. Understanding the tolerance to noise of specific quantum algorithms run on specific devices is important for determining the feasibility of quantum computing in the current noisy intermediate-scale quantum era. Of particular interest is understanding the noise sensitivity of these devices as more qubits are added to the system. Classical simulations are a useful tool to understand the effects of this noise, but direct classical simulations of open quantum systems are burdened by an exponentially growing cost in the number of qubits and a large local Hilbert space dimension. For onedimensional, shallow circuits, using tensor networks can replace this exponential cost with a linear one and simulate far wider systems than what would normally be available. In this paper, we describe a tensor network simulation of a neutral atom quantum system under the presence of noise, while introducing a purity-preserving truncation technique that compromises between the simplicity of the matrix product state and the positivity of the matrix product density operator. We apply this simulation to a near-optimized iteration of the quantum approximate optimization algorithm on a transverse field Ising model in order to investigate the influence of large system sizes on the performance of the algorithm. We find that while circuits with a large number of qubits fail more often under noise that depletes the qubit population, their outputs on a successful measurement are just as robust under Rydberg atom dissipation or qubit dephasing as smaller systems. However, such circuits might not perform as well under coherent multiqubit errors such as Rydberg atom crosstalk. We also find that the optimized parameters are especially robust to noise, suggesting that a noisier quantum system can be used to find the optimal parameters before switching to a cleaner system for measurements of observables.

Allen, James↗
NASA NTRS2011

Modification of Jupiter's Stratosphere Three Weeks After the 2009 Impact

Infrared spectroscopy sensitive to thermal emission from Jupiter's stratosphere reveals effects persisting 23 days after the impact of a body in late July 2009. Measurements obtained on 2009 August II UT at the impact latitude of 56 S (planetocentric), using the Goddard Heterodyne Instrument for Planetary Wind and Composition mounted on the NASA Infrared Telescope Facility, reveal increased ethane abundance and the effects of aerosol opacity. An interval of reduced thermal continuum emission at 11. 744 ~lm is measured ~60o-80 towards planetary east of the impact site, estimated to be at 3050 longitude (System Ill). Retrieved stratospheric ethane mole fraction in the near vicinity of the impact site is enhanced by up to -60% relative to quiescent regions at this latitude. Thermal continuum emission at the impact site, and somewhat west of it, is significantly enhanced in the same spectra that retrieve enhanced ethane mole fraction. Assuming that the enhanced continuum brightness near the impact site results from thermalized aerosol debris blocking contribution from the continuum formed in the upper troposphere and indicating the local temperature, then continuum emission by a haze layer can be approximated by an opaque surface inserted at the 45-60 mbar pressure level in the stratosphere in an unperturbed thermal profile, setting an upper limit on the pressure and therefore a lower limit on the altitude of the top of the impact debris at this time. The reduced continuum brightness east of the impact site can be modeled by an opaque surface near the cold tropopause, which is consistent with a lower altitude of ejecta/impactor-formed opacity or significantly lesser column density of opaque haze material. The physical extent of the observed region of reduced continuum implies a minimum average velocity of 21 m/s transporting material prograde (planetary east) from the impact.

Fast, Kelly E.↗
NASA NTRS2022

Estimating groundwater use and demand in arid Kenya through assimilation of satellite data and in-situ sensors with machine learning toward drought early action

Groundwater is an important source of water for people, livestock, and agriculture during drought in the Horn of Africa. In this work, areas of high groundwater use and demand in drought-prone Kenya were identified and forecasted prior to the dry season. Estimates of groundwater use were extended from a sentinel network of 69 in-situ sensored mechanical boreholes to the region with satellite data and a machine learning model. The sensors contributed 756 site-month observations from June 2017 to September 2021 for model building and validation at a density of approximately one sensor per 3700 sq.km. An ensemble of 19 parameterized algorithms was informed by features including satellite-derived precipitation, surface water availability, vegetation indices, hydrologic land surface modeling, and site characteristics to dichotomize high groundwater pump utilization. Three operational definitions of high demand on groundwater infrastructure were considered: 1) mechanical runtime of pumps greater than a quarter of a day (6+ hr) and daily per capita volume extractions indicative of 2) domestic water needs (35+ L), and 3) intermediate needs including livestock (75+ L). Gridded interpolation of localized groundwater use and demand was provided from 2017 to 2020 and forecasted for the 2021 dry season, June–September 2021. Cross-validated skill for contemporary estimates of daily pump runtime and daily volume extraction to meet domestic and intermediate water needs was 68%, 69%, and 75%, respectively. Forecasts were externally validated with an accuracy of at least 56%, 70%, or 72% for each groundwater use definition. The groundwater maps are accessible to stakeholders including the Kenya National Drought Management Authority (NDMA) and the Famine Early Warning Systems Network (FEWS NET). These maps represent the first operational spatially-explicit sub-seasonal to seasonal (S2S) estimates of groundwater use and demand in the literature. Knowledge of historical and forecasted groundwater use is anticipated to improve decision-making and resource allocation for a range of early warning early action applications.

Katie Fankhauser↗
DOE OSTI2023

Accelerating self-consistent field iterations in Kohn-Sham density functional theory using a low-rank approximation of the dielectric matrix

We present an efficient preconditioning technique for accelerating the fixed-point iteration in real-space Kohn-Sham density functional theory (DFT) calculations. The preconditioner uses a low-rank approximation of the dielectric matrix (LRDM) based on Gâteaux derivatives of the residual of fixed-point iteration along appropriately chosen direction functions. We develop a computationally efficient method to evaluate these Gâteaux derivatives in conjunction with the Chebyshev filtered subspace iteration procedure, an approach widely used in large-scale Kohn-Sham DFT calculations. Further, we propose a variant of LRDM preconditioner based on adaptive accumulation of low-rank approximations from previous self-consistent field iterations, and also extend the LRDM preconditioner to spin-polarized Kohn-Sham DFT calculations. We demonstrate the robustness and efficiency of the LRDM preconditioner against other widely used preconditioners on a range of benchmark systems with sizes ranging from ~100 to 1100 atoms (~500–20,000 electrons). The benchmark systems include various combinations of metal-insulating-semiconducting heterogeneous material systems, nanoparticles with localized d orbitals near the Fermi energy, nanofilm with metal dopants, and magnetic systems. In all benchmark systems, the LRDM preconditioner converges robustly within 20–30 iterations. In contrast, other widely used preconditioners show slow convergence in many cases, as well as divergence of the fixed-point iteration in some cases. Lastly, we demonstrate the computational efficiency afforded by the LRDM method, with up to 3.4-fold reduction in computational cost for the total ground-state calculation compared to other preconditioners.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS1979

Stark-profile calculations for Lyman-series lines of one-electron ions in dense plasmas

The frequency distributions of the first six Lyman lines of hydrogenlike carbon, oxygen, neon, magnesium, aluminum, and silicon ions broadened by the local fields of both ions and electrons are calculated for dense plasmas. The electron collisions are treated by an impact theory allowing (approximately) for level splittings caused by the ion fields, finite duration of the collisions, and screening of the electron fields. These calculations are fully quantum mechanical and include the full Coulomb interaction. Ion effects are calculated in the quasistatic, linear-Stark-effect approximation, using distribution functions of Hooper and Tighe which include correlation and shielding effects. Theoretical uncertainties from the various approximations are estimated, and the scaling of the profiles with density, temperature, and nuclear charge is discussed. A correction for the effects caused by low-frequency field fluctuations is suggested.

Griem, H. R.↗
NASA NTRS1983

Evidence for temporal variations in polar plumes

The variability of the O VI and Mg X emissions from four plumes is studied in order to place empirical constraints on possible plasma heating processes and acceleration mechanisms for mass flows in polar plumes. The observations comprise a 40 min seqeunce of spectroheliograms acquired with a temporal resolution of approximately 2 min. It is found that the Mg X emission from each plume has a strong 'dc' component and that any variations occurred at approximately the 10 percent (of the mean) level or less. This means that temporal variations in the density and/or temperature were 5 percent or less. Some statistically significant variations in the measured Mg X signals are seen. It is noted that two of the plumes decreased in brightness by approximately 10 percent during the 40 min period of the observations. Besides this, two plumes exhibited short-term (few minute) variations that are seen as statistically significant at a confidence level of more than 90 percent. The latter variations (if real) are thought to be due to small (approximately 20,000 K) fluctuations in temperature caused by fluctuations in the local plasma heating rate.

Withbroe, G. L.↗
DOE OSTI2021

Pellet triggering of edge localized modes in low collisionality pedestals at DIII-D

Edge localized modes (ELMs) are triggered using deuterium pellets injected into plasmas with ITER-relevant low collisionality pedestals, and the resulting peak ELM energy fluence is reduced by approximately 25%–50% relative to natural ELMs destabilized at similar pedestal pressures. Cryogenically frozen deuterium pellets are injected from the low-field side of the DIII-D tokamak at frequencies lower than the natural ELM frequency, and heat flux is measured by infrared cameras. Ideal MHD pedestal stability calculations show that without pellet injection, these low collisionality pedestals were limited by their current density (peeling-limited) rather than their pressure gradient (ballooning-limited). ELM triggering success correlates strongly with pellet mass, consistent with the theory that a large pressure perturbation is required to trigger an ELM in low collisionality discharges that are far from the ballooning stability boundary. For sufficiently large pellets, both instantaneous and time-integrated ELM energy deposition measured by infrared cameras is reduced with respect to naturally occurring ELMs at the inner strike point, which is the position where it is largest for natural ELMs. Energy fluence at the outer strike point is less effected. Cameras observing both heat flux and D-alpha emission often find significant toroidally asymmetric striations in the outboard far scrape-off layer resulting from ELMs that are triggered by pellets. Toroidal asymmetries at the inner strike point are similar between natural and pellet-triggered ELMs, indicating that the reduction in peak heat flux and total fluence at that location is robust for the conditions reported here.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Electronic correlation induced expansion of Fermi pockets in δ-plutonium

Plutonium is a critically important material as the behavior of its 5f electrons stands midway between the metalliclike itinerant character of the light actinides and localized atomic-core-like character of the heavy actinides. The δ phase of plutonium (δ-Pu), whereas still itinerant, has a large coherent Kondo peak and strong electronic correlations coming from its near-localized character. Here, using sophisticated Gutzwiller wave function and dynamical mean-field theory correlated theories, we study the Fermi surface and associated mass renormalizations of δ-Pu together with calculations of the de Haas–van Alphen frequencies. We find a large (~200%) correlation induced volume expansion in both the hole and the electron pockets of the Fermi surface in addition to an intermediate mass enhancement. All of the correlated electron theories predict, approximately, the same hole pocket placement in the Brillouin zone, which is different from that obtained in conventional density-functional band-structure theory, whereas the electron pockets from all theories are in, roughly, the same place.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗